derivative (x^3-2x)(x^{-4}+x^{-2})

Lösung

ableitung von

(x^{3} - 2 x) (x^{- 4} + x^{- 2})

\frac{1}{x ^{2}} + \frac{6}{x ^{4}} + 1

Schritte zur Lösung

\frac{d}{d x} ((x^{3} - 2 x) (x^{- 4} + x^{- 2}))

Wende die Produktregel an:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x^{3} - 2 x, g = x^{- 4} + x^{- 2}

= \frac{d}{d x} (x^{3} - 2 x) (x^{- 4} + x^{- 2}) + \frac{d}{d x} (x^{- 4} + x^{- 2}) (x^{3} - 2 x)

\frac{d}{d x} (x^{3} - 2 x) = 3 x^{2} - 2

\frac{d}{d x} (x^{- 4} + x^{- 2}) = - \frac{4}{x ^{5}} - \frac{2}{x ^{3}}

= (3 x^{2} - 2) (x^{- 4} + x^{- 2}) + (- \frac{4}{x ^{5}} - \frac{2}{x ^{3}}) (x^{3} - 2 x)

Vereinfache

(3 x^{2} - 2) (x^{- 4} + x^{- 2}) + (- \frac{4}{x ^{5}} - \frac{2}{x ^{3}}) (x^{3} - 2 x) : \frac{1}{x ^{2}} + \frac{6}{x ^{4}} + 1

= \frac{1}{x ^{2}} + \frac{6}{x ^{4}} + 1

t a y l or \frac{2}{( 3 x + 1 )}, 2

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\frac{d}{d t} (t^{3} - 12 t^{2} + 45 t)